Uncountable Set less Countable Set is Uncountable

Theorem

Let $S$ be an uncountable set.

Let $T \subseteq S$ be a countable subset of $S$.

Then:

$S \setminus T$ is uncountable

where $\setminus$ denotes set difference.

Proof

Aiming for a contradiction, suppose $S \setminus T$ were countable.

By definition of relative complement:

$S \setminus T = \relcomp S T$

Thus from Union with Relative Complement:

$\paren {S \setminus T} \cup T = S$

But from Finite Union of Countable Sets is Countable it follows that $S$ is countable.

From this contradiction it follows that $S \setminus T$ is uncountable.

$\blacksquare$

Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle, by way of Union with Relative Complement.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom.

This in turn invalidates this theorem from an intuitionistic perspective.